10 The Aharonov–Bohm effect

Consider once more the two-slit experiment with electrons. What happens if we place a thin, long solenoid right behind (or right in front of) the slit plate, between the two slits. If no current is flowing in the solenoid, we observe the usual interference pattern. But if a current is flowing, the interference pattern is shifted sideways (right or left, depending on the direction of the current).

This effect could easily have been predicted in the late 1920’s, by which time all the necessary physics was in place. Yet by many accounts it was predicted three decades later, in 1959, by Yakir Aharonov and David Bohm.[1] (Actually it was first predicted two decades later,[2] but it made a splash only after the publication of the paper by Aharonov and Bohm. This was followed by an actual demonstration of the effect in less than a year.) Why did it take that long for this effect to be predicted and taken note of?

Take another look at the rectangle on the right side of Figure 2.17.1. The flux of the mag­netic field through this rec­tan­gle, we observed, deter­mines the dif­fer­ence between the actions of the paths A→B→C and A→D→C. Think of these paths as leading from the electron gun G through either L or R to a detector D at the screen: G→L→D and G→R→D. As we glean from Equation 2.17.1, the action for each path depends on the vector potential along the path. The vector potential A (as well as the magnetic field B) associated with a current-bearing solenoid are shown in Figure 3.10.1. If this solenoid is introduced into the setup of the two-slit experiment, the action for the left path (G→L→D) is increased, while the action for the right path (G→R→D) is reduced. Stepping up the current in the solenoid increases the difference between these actions. Since this difference determines the positions of the maxima and minima at the screen, the interference pattern is shifted to the left as a result.

A solenoid, the current J flowing through it, and the resulting fields A and B.

There are several reasons why it took that long to predict this remarkable effect. For one, classical electrodynamics can be formulated exclusively in terms of the electric field E and the magnetic field B. For another, while the four components of V and A uniquely determine the six components of E and B, they themselves are not not unique. E and B are invariant under the substitutions

(3.10.1)xxxV → V−∂tα,xxxAx → Ax+∂xα,xxxAy → Ay+∂yα,xxxAz → Az+∂zα,

where α is a func­tion of the space­time coor­di­nates t,x,y,z.

On the other hand, it was well known that the Schrödinger equation could accommodate electromagnetic effects only in terms of the potentials V and A, and that the actions associated with loops are also invariant under these substitutions. (Since these substitution change the actions for G→L→D and G→R→D by equal amounts, the difference between the two actions, which equals the action associated with the loop G→L→D→R→G, remains unchanged.) Yet physicists were unable to disabuse themselves of the notion that E and B are physically real, while the potentials “have no physical meaning and are introduced solely for the purpose of mathematical simplification of the equations,” as Fritz Rohrlich wrote.[3] The general idea at the time was that the electromagnetic field is a physical entity in its own right, that it is locally acted upon by charges, that it locally acts on charges, and that it mediates the action of charges on charges by locally acting on itself. At the heart of this notion was the so-called principle of local action, felicitously articulated by DeWitt and Graham[4] in an American Journal of Physics resource letter:

physicists are, at bottom, a naive breed, forever trying to come to terms with the “world out there” by methods which, however imaginative and refined, involve in essence the same element of contact as a well-placed kick.

With the notable exception of Roger Boscovich, a Croatian physicist and philosopher who flourished in the 18th Century, it does not seem to have occurred to anyone that local action is as unintelligible as the apparent ability of material objects to act where they are not, which the principle of local action has supposedly done away with. (The impression that local action is intelligible rests on such familiar experiences as pulling a rope or pushing a stalled car. As soon as we take a microscopic look at what happens between the pushing hands and the car, we discover that this seemingly local action involves net interatomic or intermolecular repulsive forces acting at a distance.)

If it is believed that electromagnetic effects on charges are produced via a continuous sequence of local cause-effect relations, then something like the Aharonov–Bohm effect cannot be foreseen, for the values of E and B along the alternative electron paths can be made arbitrarily small by making the solenoid sufficiently long. And even if A were a physical entity and as such responsible for the Aharonov–Bohm effect, it could not produce it locally since the local values of A are arbitrary.